Locally compactness characterizes finite dimensionality.
Recall that a TVS is said to be locally compact if $0$ has a compact neighborhood. For a Hausdorff TVS $V$,
\[\dim(V)<\infty \Leftrightarrow V \text{ is locally compact}.\]See a proof here.
Example:
In the space $C[0,1]$ with the sup norm, consider $\{ f_n(x) \}$, where $f_n$ vanishes outside the interval $I_n=[\frac{1}{n+1}, \frac{1}{n}]$, takes the value $1$ at the midpoint of $I_n$, and is linearly interpolated within $I_n$. Then $\Vert f_n \Vert=1$ for all $n$ and $\Vert f_n-f_m\Vert=1$ whenever $n\neq m$. This implies the unit ball is not compact.
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Topology